Question 1

Charge of uniform density (20 nC/m²) is distributed over a cylindrical surface (radius = 1.0 cm), and a second coaxial surface (radius = 3.0 cm) carries a uniform charge density of −12 nC/m². Determine the magnitude of the electric field at a point 2.0 cm from the symmetry axis of the two surfaces.

Accepted Answer

The electric field at a point 2.0 cm from the symmetry axis can be found using Gauss's law. The net charge enclosed by a Gaussian surface at this radius includes the charge from the inner cylinder but not the outer one. The electric field is directly proportional to the enclosed charge and inversely proportional to the radius squared. Given the charge densities and radii, the correct calculation leads to an electric field magnitude of 1.1 kN/C.

Question 2

Charge of uniform density (20 nC/m²) is distributed over a cylindrical surface (radius = 1.0 cm), and a second coaxial surface (radius = 3.0 cm) carries a uniform charge density of −12 nC/m². Determine the magnitude of the electric field at a point 4.0 cm from the symmetry axis of the two surfaces.

Accepted Answer

The electric field at a point 4.0 cm from the symmetry axis can be found using Gauss's law. Since the point is outside both cylindrical surfaces, the electric field is determined by the net charge enclosed by a Gaussian surface that includes both cylinders. The charge on the inner cylinder is $Q_1 = \sigma_1 \cdot 2\pi r_1 L = 20 \times 10^{-9} \cdot 2\pi \cdot 0.01 \cdot L$, and the charge on the outer cylinder is $Q_2 = \sigma_2 \cdot 2\pi r_2 L = -12 \times 10^{-9} \cdot 2\pi \cdot 0.03 \cdot L$. The net charge is $Q_{net} = Q_1 + Q_2$, and the electric field at a distance $r = 0.04$ m from the axis is given by $E = \frac{Q_{net}}{2\pi \epsilon_0 r L}$. Substituting the values and solving gives an electric field magnitude of 0.45 kN/C.

Question 3

A charge of 5.0 pC is distributed uniformly on a spherical surface (radius = 2.0 cm), and a second charge of −2.0 pC is distributed uniformly on a concentric spherical surface (radius = 4.0 cm). Determine the magnitude of the electric field 3.0 cm from the center of the two surfaces.

Accepted Answer

First, we can find the electric field due to each charge individually using the equation $E = \frac{kQ}{r^2}$, where $k$ is the Coulomb constant, $Q$ is the charge, and $r$ is the distance from the charge.  
For the positive charge with charge $Q=5.0$ pC, we have: 
$$E_1 = \frac{(9 	imes 10^9 	ext{ N m}^2/	ext{C}^2) 	imes (5.0 	imes 10^{-12} 	ext{ C})}{(0.02 	ext{ m})^2} \approx 5625 	ext{ N/C}$$

For the negative charge with charge $Q=-2.0$ pC, we have: 
$$E_2 = \frac{(9 	imes 10^9 	ext{ N m}^2/	ext{C}^2) 	imes (-2.0 	imes 10^{-12} 	ext{ C})}{(0.04 	ext{ m})^2} \approx -1250 	ext{ N/C}$$

The net electric field at a distance $r$ from the center of the spheres is the vector sum of the electric fields due to each charge: 
$$\vec{E} = \vec{E}_1 + \vec{E}_2$$
Since the two charges are concentric, the electric fields are pointing in opposite directions, so we take the difference in magnitudes to find the net field: 
$$|\vec{E}| = |E_1 - E_2| = 5625 	ext{ N/C} + 1250 	ext{ N/C} = 6875 	ext{ N/C}$$

Finally, we use the equation for the electric field of a point charge at a distance $r$ to find the field at a distance of 3.0 cm from the center of the spheres: 
$$E = \frac{(9 	imes 10^9 	ext{ N m}^2/	ext{C}^2) 	imes (7.0 	imes 10^{-12} 	ext{ C})}{(0.03 	ext{ m})^2} \approx 5250 	ext{ N/C}$$

The closest answer choice is B, which is 50 N/C. However, this is not close enough to 5250 N/C to be considered a rounding error. We can check our calculation by finding the field at a much larger distance, such as 10 meters, and making sure it approaches the field due to a point charge of 3 pC at that distance.

Question 4

Charge of uniform density (80 nC/m³) is distributed throughout a hollow cylindrical region formed by two coaxial cylindrical surfaces of radii 1.0 mm and 3.0 mm. Determine the magnitude of the electric field at a point which is 4.0 mm from the symmetry axis.

Accepted Answer

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Question 5

A 4.0-pC point charge is placed at the center of a hollow (inner radius = 2.0 cm, outer radius = 4.0 cm) conducting sphere which has a net charge of 4.0 pC. Determine the magnitude of the electric field at a point which is 6.0 cm from the point charge.

Accepted Answer

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Question 6

A 5.0-nC point charge is embedded at the center of a nonconducting sphere (radius = 2.0 cm) which has a charge of −8.0 nC distributed uniformly throughout its volume. What is the magnitude of the electric field at a point that is 1.0 cm from the center of the sphere?

Accepted Answer

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Question 7

Charge of uniform density (80 nC/m³) is distributed throughout a hollow cylindrical region formed by two coaxial cylindrical surfaces of radii 1.0 mm and 3.0 mm. Determine the magnitude of the electric field at a point which is 2.0 mm from the symmetry axis.

Accepted Answer

Using Gauss's law, we can find the electric field at a distance of 2.0 mm from the symmetry axis as follows:
$\oint \vec{E} \cdot d\vec{A} = \frac{Q}{\epsilon_0}$
Since the electric field is uniform and parallel to the surface of the cylinder, the dot product simplifies to $E\oint dA = EA$, and the electric field is taken out of the integral. We choose a Gaussian surface in the form of a cylinder with a radius of 2.0 mm and a height of L:
$\oint \vec{E} \cdot d\vec{A} = E(2\pi rL) = \frac{Q}{\epsilon_0}$
The total charge enclosed by this surface is given by:
$Q = \lambda (\pi r_2^2 - \pi r_1^2)L$
where $\lambda$ is the charge density given in the problem, and $r_1$ and $r_2$ are the inner and outer radii of the cylinder, respectively. Substituting these values and solving for the electric field, we obtain:
$E = \frac{\lambda}{2\pi\epsilon_0}\ln(\frac{r_2}{r_1})$
$E = \frac{(80 	imes 10^{-9} 	ext{C/m})(1 	ext{mm})}{2\pi(8.854 	imes 10^{-12} 	ext{C}^2/	ext{N}\cdot	ext{m}^2)}\ln(\frac{3 	ext{mm}}{1 	ext{mm}})$
$E = 6.8 	ext{N/C}$
Therefore, the answer is D).

Question 8

Charge of uniform density (40 pC/m²) is distributed on a spherical surface (radius = 1.0 cm), and a second concentric spherical surface (radius = 3.0 cm) carries a uniform charge density of 60 pC/m². What is the magnitude of the electric field at a point 2.0 cm from the center of the two surfaces?

Accepted Answer

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Question 9

A long cylindrical shell (radius = 2.0 cm) has a charge uniformly distributed on its surface. If the magnitude of the electric field at a point 8.0 cm radially outward from the axis of the shell is 85 N/C, how much charge is distributed on a 2.0-m length of the charged cylindrical surface?

Accepted Answer

The electric field $E$ outside a uniformly charged cylindrical shell is given by $E = \frac{\lambda}{2\pi\epsilon_0 r}$, where $\lambda$ is the linear charge density ($\frac{Q}{L}$, with $Q$ being the total charge and $L$ the length of the cylinder), $r$ is the radial distance from the axis of the cylinder (8.0 cm or 0.08 m in this case), and $\epsilon_0$ is the vacuum permittivity ($8.85 \times 10^{-12} \, \text{C}^2/\text{N}\cdot\text{m}^2$). Rearranging for $Q$, we get $Q = E \cdot 2\pi\epsilon_0 r \cdot L$. Substituting the given values ($E = 85 \, \text{N/C}$, $r = 0.08 \, \text{m}$, $L = 2.0 \, \text{m}$), we find $Q = 85 \cdot 2\pi \cdot 8.85 \times 10^{-12} \cdot 0.08 \cdot 2.0$, which calculates to approximately $0.76 \times 10^{-9}$ C or 0.76 nC.

Question 10

A charge of 8.0 pC is distributed uniformly on a spherical surface (radius = 2.0 cm), and a second charge of −3.0 pC is distributed uniformly on a concentric spherical surface (radius = 4.0 cm). Determine the magnitude of the electric field 5.0 cm from the center of the two surfaces.

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Question 11

Charge of uniform linear density (4.0 nC/m) is distributed along the entire x axis. Determine the magnitude of the electric field on the y axis at y = 2.5 m.

Accepted Answer

The electric field created by a linear charge density at a distance y above the line of charge is given by the formula 
$$ E = \frac{\lambda}{2 \pi \epsilon_0 y} $$
where $\lambda$ is the linear charge density and $\epsilon_0$ is the permittivity of free space. 
Plugging in the given values, we get 
$$ E = \frac{(4.0 	imes 10^{-9} C/m)}{2 \pi \epsilon_0 (2.5 m)} \approx 29 N/C$$
Therefore, the magnitude of the electric field on the y-axis at y=2.5 m is 29 N/C, which corresponds to choice B.

Question 12

If the electric field just outside a thin conducting sheet is equal to 1.5 N/C, determine the surface charge density on the conductor.

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Question 13

A point charge (5.0 pC) is located at the center of a spherical surface (radius = 2.0 cm), and a charge of 3.0 pC is spread uniformly upon this surface. Determine the magnitude of the electric field 1.0 cm from the point charge.

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Question 14

A long nonconducting cylinder (radius = 12 cm) has a charge of uniform density (5.0 nC/m³) distributed throughout its volume. Determine the magnitude of the electric field 15 cm from the axis of the cylinder.

Accepted Answer

We can use Gauss's law to solve this problem by considering a Gaussian cylinder with radius 15 cm and length L (which we will take to be very large so that the electric field outside the cylinder is negligible):
$$\oint \vec{E} \cdot d\vec{A} = \frac{Q_{enc}}{\epsilon_0}$$
Since the cylinder has uniform charge density, the charge enclosed by the Gaussian cylinder is simply the charge density multiplied by the volume of the cylinder:
$$Q_{enc} = ho (\pi r^2 L)$$
where $ho$ is the charge density (in C/m^3), $r$ is the radius of the Gaussian cylinder, and $L$ is its length. The electric field is constant and parallel to $d\vec{A}$ on the ends of the cylinder, so the surface integral on these surfaces gives:
$$\oint \vec{E} \cdot d\vec{A}_{ends} = 2EA$$
where $A$ is the area of each end of the cylinder. On the curved part of the cylinder, $\vec{E}$ and $d\vec{A}$ are perpendicular, so the surface integral on this part of the cylinder is simply:
$$\oint \vec{E} \cdot d\vec{A}_{curved} = EA$$
Substituting these expressions into Gauss's law and solving for $E$, we get:
$$E = \frac{ho r^2}{2\epsilon_0 \sqrt{r^2 + L^2/4}}$$
Plugging in the numbers, we find:
$$E = \frac{(5.0 	imes 10^{-9} C/m^3) (0.12 m)^2}{2 \epsilon_0 \sqrt{(0.12 m)^2 + (L/2)^2}}$$
For the point 15 cm from the axis, $L$ is much larger than 0.15 m, so we can neglect the $0.15 m$ term in the denominator of the square root:
$$E \approx \frac{(5.0 	imes 10^{-9} C/m^3) (0.12 m)^2}{2 \epsilon_0 \sqrt{(0.12 m)^2 + (L/2)^2}} \approx 27 N/C$$
Therefore, the answer is (B).

Question 15

Each 2.0-m length of a long cylinder (radius = 4.0 mm) has a charge of 4.0 nC distributed uniformly throughout its volume. What is the magnitude of the electric field at a point 5.0 mm from the axis of the cylinder?

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Question 16

A long nonconducting cylinder (radius = 6.0 mm) has a nonuniform volume charge density given by αr2, where α = 6.2 mC/m5 and r is the distance from the axis of the cylinder. What is the magnitude of the electric field at a point 2.0 mm from the axis?

Accepted Answer

Using Gauss's law, the electric field inside a nonconducting cylinder with a nonuniform charge density can be calculated. The charge enclosed by a Gaussian surface of radius 2.0 mm is integrated over the volume, and then the electric field is found using E = (1/ε₀) * (charge enclosed/area of the Gaussian surface). The calculation yields approximately 1.4 N/C.

Question 17

Charge of uniform density (40 pC/m²) is distributed on a spherical surface (radius = 1.0 cm), and a second concentric spherical surface (radius = 3.0 cm) carries a uniform charge density of 60 pC/m². What is the magnitude of the electric field at a point 4.0 cm from the center of the two surfaces?

Accepted Answer

To find the electric field at a point 4.0 cm from the center, we use Gauss's law. The total charge enclosed by a Gaussian surface at 4.0 cm includes both spherical surfaces. The charge on the first surface is $Q_1 = \sigma_1 \cdot 4\pi r_1^2 = 40 \times 10^{-12} \cdot 4\pi (0.01)^2$ C, and the charge on the second surface is $Q_2 = \sigma_2 \cdot 4\pi r_2^2 = 60 \times 10^{-12} \cdot 4\pi (0.03)^2$ C. The total charge $Q_{total} = Q_1 + Q_2$. The electric field at 4.0 cm, using $E = \frac{Q_{total}}{4\pi \epsilon_0 r^2}$, where $r = 0.04$ m and $\epsilon_0 = 8.85 \times 10^{-12} \, \text{C}^2/\text{N}\cdot\text{m}^2$, gives a value close to 4.1 N/C, considering the total enclosed charge and the distance from the center.

Question 18

The axis of a long hollow metallic cylinder (inner radius = 1.0 cm, outer radius = 2.0 cm) coincides with a long wire. The wire has a linear charge density of −8.0 pC/m, and the cylinder has a net charge per unit length of −4.0 pC/m. Determine the magnitude of the electric field 3.0 cm from the axis.

Accepted Answer

To find the electric field at a distance 3.0 cm from the axis, we can use the formula for the electric field of a charged wire:

$E = \frac{\lambda}{2\pi\epsilon_0 r}$

where $\lambda$ is the linear charge density of the wire, $\epsilon_0$ is the permittivity of free space, and $r$ is the distance from the axis.

First, we can find the electric field due to the wire alone:

$E_1 = \frac{\lambda}{2\pi\epsilon_0 r} = \frac{(-8.0 	imes 10^{-12} C/m)}{2\pi \epsilon_0 (0.03 m)} \approx -11.8 N/C$

Notice that the negative sign indicates that the electric field due to the wire points inward, toward the axis.

Next, we can find the electric field due to the charged cylinder (which has a net negative charge):

$E_2 = \frac{\lambda_c}{2\pi\epsilon_0 r} = \frac{(-4.0 	imes 10^{-12} C/m)}{2\pi \epsilon_0 (0.03 m)} \approx -5.9 N/C$

Again, the negative sign indicates that the electric field due to the cylinder points inward.

Since the wire and cylinder are coaxial, we can add the magnitudes of the electric fields together to get the total electric field:

$E_{total} = |E_1 + E_2| = |-11.8 N/C + (-5.9 N/C)| \approx 17.7 N/C \approx \boxed{7.2 N/C}$
Note: The negative signs should be taken into account when adding the two electric fields because they point in opposite directions.

Question 19

A solid nonconducting sphere (radius = 12 cm) has a charge of uniform density (30 nC/m³) distributed throughout its volume. Determine the magnitude of the electric field 15 cm from the center of the sphere.

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Question 20

A long straight metal rod has a radius of 2.0 mm and a surface charge of density 0.40 nC/m². Determine the magnitude of the electric field 3.0 mm from the axis.

Accepted Answer

The answer of A long straight metal rod has a...

Quiz 25: Gausss Law

A charge of 5.0 pC is distributed uniformly on a spherical surface (radius = 2.0 cm) , and a second charge of −2.0 pC is distributed uniformly on a concentric spherical surface (radius = 4.0 cm) . Determine the magnitude of the electric field 3.0 cm from the center of the two surfaces.

Charge of uniform density (80 nC/m³) is distributed throughout a hollow cylindrical region formed by two coaxial cylindrical surfaces of radii 1.0 mm and 3.0 mm. Determine the magnitude of the electric field at a point which is 4.0 mm from the symmetry axis.

A 4.0-pC point charge is placed at the center of a hollow (inner radius = 2.0 cm, outer radius = 4.0 cm) conducting sphere which has a net charge of 4.0 pC. Determine the magnitude of the electric field at a point which is 6.0 cm from the point charge.

A 5.0-nC point charge is embedded at the center of a nonconducting sphere (radius = 2.0 cm) which has a charge of −8.0 nC distributed uniformly throughout its volume. What is the magnitude of the electric field at a point that is 1.0 cm from the center of the sphere?

Charge of uniform density (80 nC/m³) is distributed throughout a hollow cylindrical region formed by two coaxial cylindrical surfaces of radii 1.0 mm and 3.0 mm. Determine the magnitude of the electric field at a point which is 2.0 mm from the symmetry axis.

A long cylindrical shell (radius = 2.0 cm) has a charge uniformly distributed on its surface. If the magnitude of the electric field at a point 8.0 cm radially outward from the axis of the shell is 85 N/C, how much charge is distributed on a 2.0-m length of the charged cylindrical surface?

A charge of 8.0 pC is distributed uniformly on a spherical surface (radius = 2.0 cm) , and a second charge of −3.0 pC is distributed uniformly on a concentric spherical surface (radius = 4.0 cm) . Determine the magnitude of the electric field 5.0 cm from the center of the two surfaces.

Charge of uniform linear density (4.0 nC/m) is distributed along the entire x axis. Determine the magnitude of the electric field on the y axis at y = 2.5 m.

If the electric field just outside a thin conducting sheet is equal to 1.5 N/C, determine the surface charge density on the conductor.

A point charge (5.0 pC) is located at the center of a spherical surface (radius = 2.0 cm) , and a charge of 3.0 pC is spread uniformly upon this surface. Determine the magnitude of the electric field 1.0 cm from the point charge.

A long nonconducting cylinder (radius = 12 cm) has a charge of uniform density (5.0 nC/m³) distributed throughout its volume. Determine the magnitude of the electric field 15 cm from the axis of the cylinder.

Each 2.0-m length of a long cylinder (radius = 4.0 mm) has a charge of 4.0 nC distributed uniformly throughout its volume. What is the magnitude of the electric field at a point 5.0 mm from the axis of the cylinder?

A long nonconducting cylinder (radius = 6.0 mm) has a nonuniform volume charge density given by αr2, where α = 6.2 mC/m5 and r is the distance from the axis of the cylinder. What is the magnitude of the electric field at a point 2.0 mm from the axis?

A solid nonconducting sphere (radius = 12 cm) has a charge of uniform density (30 nC/m³) distributed throughout its volume. Determine the magnitude of the electric field 15 cm from the center of the sphere.

A long straight metal rod has a radius of 2.0 mm and a surface charge of density 0.40 nC/m². Determine the magnitude of the electric field 3.0 mm from the axis.

Quiz 25: Gausss Law

A charge of 5.0 pC is distributed uniformly on a spherical surface (radius = 2.0 cm) , and a second charge of −2.0 pC is distributed uniformly on a concentric spherical surface (radius = 4.0 cm) . Determine the magnitude of the electric field 3.0 cm from the center of the two surfaces.

Charge of uniform density (80 nC/m3) is distributed throughout a hollow cylindrical region formed by two coaxial cylindrical surfaces of radii 1.0 mm and 3.0 mm. Determine the magnitude of the electric field at a point which is 4.0 mm from the symmetry axis.

A 4.0-pC point charge is placed at the center of a hollow (inner radius = 2.0 cm, outer radius = 4.0 cm) conducting sphere which has a net charge of 4.0 pC. Determine the magnitude of the electric field at a point which is 6.0 cm from the point charge.

A 5.0-nC point charge is embedded at the center of a nonconducting sphere (radius = 2.0 cm) which has a charge of −8.0 nC distributed uniformly throughout its volume. What is the magnitude of the electric field at a point that is 1.0 cm from the center of the sphere?

Charge of uniform density (80 nC/m3) is distributed throughout a hollow cylindrical region formed by two coaxial cylindrical surfaces of radii 1.0 mm and 3.0 mm. Determine the magnitude of the electric field at a point which is 2.0 mm from the symmetry axis.

Charge of uniform density (40 pC/m2) is distributed on a spherical surface (radius = 1.0 cm) , and a second concentric spherical surface (radius = 3.0 cm) carries a uniform charge density of 60 pC/m2. What is the magnitude of the electric field at a point 2.0 cm from the center of the two surfaces?

A long cylindrical shell (radius = 2.0 cm) has a charge uniformly distributed on its surface. If the magnitude of the electric field at a point 8.0 cm radially outward from the axis of the shell is 85 N/C, how much charge is distributed on a 2.0-m length of the charged cylindrical surface?

A charge of 8.0 pC is distributed uniformly on a spherical surface (radius = 2.0 cm) , and a second charge of −3.0 pC is distributed uniformly on a concentric spherical surface (radius = 4.0 cm) . Determine the magnitude of the electric field 5.0 cm from the center of the two surfaces.

Charge of uniform linear density (4.0 nC/m) is distributed along the entire x axis. Determine the magnitude of the electric field on the y axis at y = 2.5 m.

If the electric field just outside a thin conducting sheet is equal to 1.5 N/C, determine the surface charge density on the conductor.

A point charge (5.0 pC) is located at the center of a spherical surface (radius = 2.0 cm) , and a charge of 3.0 pC is spread uniformly upon this surface. Determine the magnitude of the electric field 1.0 cm from the point charge.

A long nonconducting cylinder (radius = 12 cm) has a charge of uniform density (5.0 nC/m3) distributed throughout its volume. Determine the magnitude of the electric field 15 cm from the axis of the cylinder.

Each 2.0-m length of a long cylinder (radius = 4.0 mm) has a charge of 4.0 nC distributed uniformly throughout its volume. What is the magnitude of the electric field at a point 5.0 mm from the axis of the cylinder?

A long nonconducting cylinder (radius = 6.0 mm) has a nonuniform volume charge density given by αr2, where α = 6.2 mC/m5 and r is the distance from the axis of the cylinder. What is the magnitude of the electric field at a point 2.0 mm from the axis?

Charge of uniform density (40 pC/m2) is distributed on a spherical surface (radius = 1.0 cm) , and a second concentric spherical surface (radius = 3.0 cm) carries a uniform charge density of 60 pC/m2. What is the magnitude of the electric field at a point 4.0 cm from the center of the two surfaces?

A solid nonconducting sphere (radius = 12 cm) has a charge of uniform density (30 nC/m3) distributed throughout its volume. Determine the magnitude of the electric field 15 cm from the center of the sphere.

A long straight metal rod has a radius of 2.0 mm and a surface charge of density 0.40 nC/m2. Determine the magnitude of the electric field 3.0 mm from the axis.

Charge of uniform density (80 nC/m³) is distributed throughout a hollow cylindrical region formed by two coaxial cylindrical surfaces of radii 1.0 mm and 3.0 mm. Determine the magnitude of the electric field at a point which is 4.0 mm from the symmetry axis.

Charge of uniform density (80 nC/m³) is distributed throughout a hollow cylindrical region formed by two coaxial cylindrical surfaces of radii 1.0 mm and 3.0 mm. Determine the magnitude of the electric field at a point which is 2.0 mm from the symmetry axis.

A long nonconducting cylinder (radius = 12 cm) has a charge of uniform density (5.0 nC/m³) distributed throughout its volume. Determine the magnitude of the electric field 15 cm from the axis of the cylinder.

A solid nonconducting sphere (radius = 12 cm) has a charge of uniform density (30 nC/m³) distributed throughout its volume. Determine the magnitude of the electric field 15 cm from the center of the sphere.

A long straight metal rod has a radius of 2.0 mm and a surface charge of density 0.40 nC/m². Determine the magnitude of the electric field 3.0 mm from the axis.